2007 06 IFToMM Dynamics

7/31/2019 2007 06 IFToMM Dynamics

http://slidepdf.com/reader/full/2007-06-iftomm-dynamics 1/6

12th IFToMM World Congress, Besançon (France), June18-21, 2007

Dynamics of over-constrained rigid and flexible multibody systems

E. Zahariev* J. Cuadrado† Institute of Mechanics, BAS University of La Coruna

Sofia, Bulgaria Ferrol, Spain

Abstract — the present paper deals with the problems of

dynamic simulation of over-constrained multibody systems

applied in different motion and power transferring mechanical

devices, self-locking mechanisms, etc. The elasticity of the links

is taken into account. Generalized Newton-Euler dynamic

equations are applied for the case of finite element discretization

of flexible links. An approach of decomposition of the

mechanisms in the singular configurations is proposed. Relative

and absolute nodal coordinates are used. The method substitutesthe kinematic constraints by elastic forces. An example of

dynamic analysis of over-constrained mechanisms is presented.

Keywords: dynamics simulation, over-constrained

mechanisms, flexibility.

I. Introduction

The theory of the constrained dynamics is very welldeveloped in the mechanics and widely applied inmultibody system motion simulation [1 – 5]. Theconstraints imposed on the systems have different natureand type. Most often in the theory of the mechanismskinematic and force constraints are regarded. Joints

connecting rigid bodies impose kinematic constraints,while if one regards the elastic nature of the contact

between the bodies the same joints could be regarded asforce constraints. Closed chains in the mechanical schemeimpose kinematic constraints that additionally diminishthe system degree of freedom (dof). Furthermore, thenotation “common constraints” is used for systems thathave common restrictions imposed on the motion of the

links. Such systems are: the plane mechanisms(restrictions for motion in the plane); sphericalmechanisms (the joint axes are crossing in a common point); the mechanism of Bennett [6], etc.

The dynamics of constraint system is presented byDifferential Algebraic Equations (DAE). Surveys of theexisting techniques for solving DAE may be found in [4,

7]. The classical method to deal with DAE is to expressthe constraint condition at acceleration level. This leads to

replacement of the original system by a system of Ordinary Differential Equations ODE. Maintaining theacceleration constraints one does not satisfy the positionand velocity constraints. Baumgarte’s stabilization [8]term is introduced to ensure exponential convergence of the constraint error to zero. The problem with this method

*E-mail: [email protected]†E-mail: [email protected]

is in selection of high gains to keep small constrainterrors. A similar approach based on penalty functions is

presented in [4, 9]. Implementation of both methods [4, 8,9] results in inclusion additional terms in the right side of the dynamic equations that could be treated as reactionforces in the joints cut but cannot be compared to theelastic forces in links.

Another group of researchers [10, 11, 12] proposed

projection techniques to maintain the constraint conditionswithout modification of the equations of motion.

Other methods are based on coordinate partitioning [13].At every step the set of the coordinates is partitioned of dependent and independent coordinates. However, a fixedset of independent coordinates may lead to dependentmatrix of the derivatives of the constraints [4, 11].

Manipulation of the dynamic equations in the singular configurations is a challenging realm of the investigations.The Augmented Lagrangian formulation proposed in [14,15] can handle redundant constraints in singular configurations. In [16] an approach for kinematic analysis

of mechanisms and their singular configurations using theMoore–Penrose pseudo-inverse matrix is applied. Eich-Soellner and Fuhrer [17] solved the problem of constraintstabilization using optimization algorithms and the pseudo-inverse matrix so derived. In [18] a projectionmethod is applied for simulation of constrained multibody

systems. Mechanisms in the vicinity of singular configurations are regarded. Friction is taken into account.

In [19] a pseudo-inverse matrix is proposed for effectivesolution of DAE and its application in singular configurations. However, special singular configurationsexist for which there is no general solution and specialmethods are to be developed taking into account the

elasticity of the links.In the paper an approach for dynamic analysis of multibody systems that provides general solution insingular configurations and self-locking position is proposed. Elasticity of the links is taken into account.Generalized Newton-Euler dynamic equations are appliedfor the case of finite element discretization of flexiblelinks. Relative and absolute nodal coordinate formulationis used. The closed kinematic chains are decomposed inopen chains substituting the kinematic constraints by

elastic forces due to elasticity of the links. Severalexamples of closed chain mechanisms in singular configuration and self-locking position are discussed.




II. Topology and kinematics of over-constrained

mechanisms

Over-constrained mechanisms are systems which dof areless than degrees of mobility. For example, if one analyze

a plain mechanism considering the regulations for thespatial mechanism you will obtain less dof (evennegative), while it is quite applicable. So planemechanisms could be also considered over-constrained,although in practice no one is thinking so. Sphericalmechanisms are also over-constrained and it could beeasily observed if the precision of the links and orientationof joint axes are not fulfilled within the prescribed

tolerances. In Fig. 1 some of most famous over-constrained mechanisms are shown.

Fig. 1. Plane, spherical and Bennett’s linkages regarded as over-constrained mechanisms

On the other hand, many closed chain mechanisms aremovable, while the topology analysis shows zero or negative dof. This is because of the kinematic parametersfor which the constraints equations are dependent. In

Fig. 2 the simplest examples of such plane mechanismsare shown.

Fig. 2. Plane mechanisms with dependent constraints

In some mechanisms the proportion between the shape

and size of the links is the reason for the increase of mechanism dof in specific positions, called singular configurations. Examples of mechanisms with closedchains in singular configuration for some basic groupsfrom the classification of Assur are presented in Fig. 3.

The different nature and behavior of the over-

constrained mechanism is the reason for the developmentof specific approaches for the kinematic and dynamic

analysis and simulation for almost every single case.Even for a single mechanism it could happen that different

approaches are to be applied during its motion. That

Fig. 3. Singular configurations for some basic groups of the Assur’s

classification

significantly slows down the effectiveness of thecomputations. But only for few cases such geometrical

considerations could be regarded. For complex plane andeven for simple space mechanisms, simple regulationscannot be discovered. For the case of singular configurations it is well know that the matrix of the

derivatives of the constraint equation system (Jacobeanmatrix) is singular. This analysis is an onerous task,causes additional branches of the algorithms and cannot be implemented effectively in the vicinity of singularities.For numerical simulation of flexible system discretizationof the continuum should be implemented and mass and

stiffness properties of the flexible bodies are to be reducedto a finite number of points called nodes. The node of a

flexible element is a free object that, in the three

dimensional space, has six degrees of freedom [20]. Thenode motion is restricted by elastic forces acting betweenthe neighbor nodes. In Figure 4, a flexible element withmany nodes is shown, where for the node with index i thecoordinates (using translations and Euler angles) of thenode coordinate system relative to the

Fig. 4. Flexible finite element node coordinate systems

absolute reference frame are pointed out, while for thesecond node (i+1) the linear and angular velocities of thenode are shown. The coordinates of node i are stored in a

6×1 matrix [ ] \

iiii 621qqqq L= , where the

notations 621 ,..., ,m ,mi =q are the elements of the

α 2

1

3

41

4

3

2

r

r

d d

ϕ

ϕ

γ

γ

α

β

β

i Z

X 0

Z 0

Y 0

iY

1iq

i X

flexible element

node i+1node i

2iq

21+i

&

eY

e X

e Z

51+i&

3iq

4iq

5iq

6iq1+iY

1+i Z

31+i&

61+i&

base

41+i

& 11+i

&

1+i X




matrix. “\” (backslash) denotes matrix transpose. Thesmall finite translations and rotations of node i are

compiled in a similar matrix i with elements

621 ,..., ,m ,mi

= . The kinematic characteristics of

motions of the nodes are mutually independent. The nodesof flexible elements have six degrees of freedom either

with respect to the element coordinate system ( eee Z Y X )

or to the absolute ( 000 Z Y X ) one and their motions could

be presented by virtual spatial joint with six dof as

described in details in [20]. But it should be made clear

difference between the coordinates iq and the small

possible motions i . The definition of the nodes as

coordinate systems allows the flexible particles of themultibody systems, similarly to the systems of rigid bodies, to be decomposed to systems of moving

coordinate systems connected by joints.All (of number a) coordinates of the system are placed

in matrix Qa . The left superscripts denote matrix

dimension, i.e.: Ai , A j ,i , Ak , j ,i are i×1, i×j and i×j×k

matrix–vector, plane and cubic matrices that, if oncedefined, could be missed. The coordinates Q are subject toconstraints that define the function of Q with respect to

generalized (of number g) coordinates qg .

The system is subject to d constraints (d = a – g), i.e.:

( ) 0Q d d === (1)

where 0d is d×1 zero matrix. The time derivatives are:

0QQQ

Qd =⋅=⋅

∂

∂= ∂

&&&

0QQQQQ

d

\ =⋅⊗+⋅=

∂∂&&&&&&

312

(2)

(3)

wherea ,d

QQ ∂∂ = and a ,a ,d

QQ 22∂∂

= are matrices of the

first and second order partial derivatives (the left

subscripts denote the differentiating variables). Notation“31 \ ⊗ ” presents matrix multiplication of three dimensional

(space) matrix. Eq. 1 defines two sets of dependent Qd

and independent qg coordinates, i.e., [ ] \

\ \ qQQ = .

The velocity equation, Eq. 2, could be transformed to [4]

qRQ && ⋅= (4)

and the time derivatives of Eq. 4 are as follows [20]:

qqRqRQ q &&&&&& ⋅⊗+⋅= ∂31 \

(5)

The matrices R and Rq∂ are computed from the partial

derivatives Q∂ andQ2∂

[20]. The equality

constraints represent the connectivity of the links in closedchains. The equation constraints of closed chains containdependent coordinates and, most often, they are the reasonfor the singularity of the Jacobean matrix. Equationconstraints of open branches could be directly transformed

with respect to the independent coordinates. Thisapproach is widely applied in constraint dynamics [1 – 4]and consists in virtual disconnections of joints that pertainto the closed chain. The entire constraint equations systemis then compiled inserting additional kinematic constraintsthat present the connectivity.

The principle proposed in the paper consists intransformation of a closed chain into open branch cutting

not the joints but the flexible links. Using the finiteelement approach allows the kinematic constraints to besubstituted by force constraints, i.e. by elastic forces in thenodes. Illustration of this approach applied for themechanisms of Fig. 3 is presented in Fig. 5.

Fig. 5. Transformation of a closed chain into open branches substitutingflexile links by elastic forces

Such transformation of a three contour six-link mechanism in Fig. 5 results in an open chain with four

branches and eight links. The elastic forces 1 f and 2 f in

Fig. 5 depend on the shape, size and stiffness of the linkscut. The first step is the transformation of the closed chaininto open branches representing the connectivity betweenthe coordinate systems of the rigid bodies and thecoordinate systems of the nodes of the flexible elements.

For example, for the four-bar mechanism in Fig. 3 and itstransformation (Fig. 5) the connectivity of the coordinate

systems is shown in Fig. 6. The coordinate systems with

Fig. 6. Connectivity of the coordinate systems of rigid links and nodes

0

1

2’

2”

3

1 f

0

1

2

3”3’

5’

5”

1 f

2 f

4

0

1 flexible link 2

30

1

2’

2”

3

1 f

1 f

01

2’ 3

2”

1 X 0

1Y

0 X

0Y ' X 2

' Y 2 " X 2

3 X " Y 2

3Y




indices 2’ and 2” correspond to the nodes of the flexible beam – link 2. The kinematic analysis of these two branches is a trivial task for every computer codegeneration program. For the next stage, the dynamic

analysis, the main initial preparations will consist in massdistribution of the flexible elements to the node coordinate

system (estimation of the mass matrix), as well as, theelastic forces (the stiffness matrix).

III. Dynamics of rigid and flexible mechanisms

Vector translationiC s of an object (point or node iC of

a body or a node of flexible element), and vector of the

small rotations i compile the matrix of the finite

displacements [ ] \ \ i

\ C i

i

s= . The coordinate

transformation matrix, i

,66

, for vector i is:

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡=

i ,

,i

i0

0q

33

33

(6)

where i ,33 is the matrix transformation of the coordinate

system i. Similarly to i and Eqs. 4, 5, the quasi

velocities and accelerations i& , i

&& are expressed with

respect to q , as well as, q& and q&& [20], i.e.:

where iR and iRq∂ are compiled from the partial

derivatives of i with respect to q [20].

For rigid body i the velocities that define its motion are

stored in 6×1 matrix [ ] \

i

\ i

\ C iC v=& of the velocity of

its centre of gravity iC and the body angular velocity. For a flexible element with n nodes the 6.n×1 matrix of the

velocities is [ ] \

\ n ,i

\ ,i

\ ,ii

&L&&&21= , where

[ ] \

\ j ,i

\ j ,i j ,i v=& , n ,..., , j 21= are 6×1 matrices of the

node velocities. The Newton – Euler equations define theinertia forces and moments loading the body. For rigid body i the 6×1 matrix of the inertia forces and moments inthe centre of gravity and relative to the inertial frame are:

⎥⎦

⎤⎢⎣

⎡⋅⋅⎥

⎦

⎤⎢⎣

⎡+⋅= ×

iC

i ,

, ,

C iC iC ii

0M

0

00MF

3

33

3333&& (9)

where

=iC M

( )

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⋅⋅

\

iii

,

,iii m ,m ,mdiag

J0

033

33

(10)

and im , iJ are the body mass and inertia tensor,

respectively. The underlined notations point out referenceto body fixed coordinate system. In [20] dense (no zeroelements) 6×6 mass matrices are regarded. These matricesare computed in cases of mass reduction, as it is for thefinite element discretization, for which the nodetranslations and rotations are dependent. For such matrices

the generalized Newton – Euler equations [20] are:

+⋅=iC iC iC

&&MF

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ⋅⋅−⎥

⎦

⎤⎢⎣

⎡⋅⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ×

××

×

0

vM

vM

v

0

3

33ii C i

iC i

C iC

ii

,i

(11)

The mass matrix of a flexible element i with n nodes is

6.n×6.n symmetric positive defined dense matrix iM .

The mass matrices are computed assuming theequivalence of the kinetic energy of the deformable particles to the energy of the masses reduced to the nodes.

If small relative deflections within the elements areassumed, these matrices are considered constant. Themass reduction is implemented on the basis of polynomial

approximation of the beam deflections [21]. The kineticenergy of such an element is:

ii \ ii

\ iii

\ ii EK &&&& ⋅⋅=⋅⋅⋅⋅= MM

2

1

2

1 (12)

where ( )iiii ,..., ,diag= is 6.n×6.n coordinate

system transformation matrix for flexible element i; iM is

6.n×6.n element mass matrix relative to inertial reference

frame. The inertia forces (6.n×1 matrix iF ) in the nodes

of flexible element i with n nodes are defined as follows:

iiiiiiii&&&&& ⋅−⋅⋅+⋅= ⊗ MMMF (13)

where, ⊗⊗⊗⊗ = n ,i ,i ,ii , , ,diag &L&&&21 ;

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

××

×⊗

j ,i j ,i

, j ,i

j ,iv

033&

is generalized skew-symmetric matrix of the linear and

angular velocities of node j from element i;

[ ] \

\ n ,i

\ ,i

\ ,ii

&L&&&21= ;

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ⋅=

×

0

v

3

j ,i j ,i j ,i

& .

The nodes of flexible elements achieve largedisplacements with respect to the inertial reference frame,

( ) ( ) qqRvqq, &&& ⋅=⎥⎦⎤⎢

⎣⎡= i

i

C i

i

( ) =⎥⎦

⎤⎢⎣

⎡=

i

C i

i

&

&&&&&&

vq,qq, ( ) ( ) qqqRqqR q &&&& ⋅⊗+⋅ ∂

31 \ ii

(7)

(8)




while the deflections relative to the element coordinatesystem are small. In order the finite element stiffnessmatrices to be correctly applied the elastic forces loadingthe nodes are to be computed using only the small relative

deflections of the nodes relative to the element.The small deflections of element i with n nodes are

compiled, similarly to the velocities in Sec. 4.1, in a 6.n×1

matrix [ ] \

\ n ,i

\ ,i

\ ,ii L21= . The element stiffness

properties are presented by 6.n×6.n sized stiffness iK .

For the finite elements the matrix iK is transformed to

the absolute reference frame to compile the stiffness

matrix iK . The elastic forces [ ] \ \ n ,i

\ ,i

\ ,ii SSSS L21=

with respect to the absolute reference frame are computed

using the relation iii ⋅−= KS . In a similar way the

elastic forces relative to the moving element coordinatesystem are calculated taking into account the regulationsfor selection the reference coordinate systems of theflexible elements [22]. For example, the elastic forces inthe beam element coordinate system are calculated by thewell known stiffness matrix using the relation [20]:

[ ] \ \ n ,i

\ ,i

,iiii L2

61 0KKS ⋅−=⋅−== (14)

The process of computation of the node elastic forces

goes trough the following steps:- transformation of the coordinate systems of the nodesrelative to the element coordinate systems;- computation of the small relative node deflections;- computation of the elastic forces in the nodes;- transformation of the elastic forces to the absolute frame.

The final form of the dynamic equations with l rigid

bodies, m flexible elements and n external forces isderived summing up the reduced inertia forces (Eqs. 11,13) for all rigid and flexible objects, as well as, for allreduced external forces including the elastic forces (with

common notationi M G , i = 1, 2, …, n), i. e.:

( )[ ]∑= ⋅+⋅⋅⋅

l

i

iiC \ C C iC

\ C ,iii

1qFRqRMR&&&

+

( )[ ]∑=

⋅+⋅⋅⋅m

i

ii \ iii

\ i ,

1

&&&& qFRqRMR

0GR gn

i

M \ M ii

=⋅−∑=1

(15)

Eg. 15 is g×1 linear system of ordinary differentialequations for the generalized accelerations.

IV. Example

An example of application of the approach proposed tomotion simulation of the six-link mechanism in Fig. 5 incase of singular configuration with no initial velocity is

presented. The mechanism is of three chains and isdecomposed of four branches. The kinematic scheme andelastic forces in the nodes of flexible element, link 3,

(' x f

3,

' y f 3 ' m3 , " m3 ,

" x f 3

," y f

3) are presented in Fig. 7.

Singular configuration for this mechanism is when the

directrixes of the link 2, 3 (nodes 3’-3”) and 5 (nodes 5’-5”) are crossing in a common point. For this case the

Jacobean matrix, as well as, the mass-matrix of thedynamic equations are singular.

Using the approach proposed no kinematic constraintsthat describe the connectivity of nodes 3’ – 3” (of link 3)and 5’ – 5” (of link 5) are applied. These constraints are

the reason for the singularity and it is avoided substitutingthem by external elastic forces. On every step the relative position of the coordinate systems of links 3’ – 3” and 5’ –

5” is estimated and the small relative node deflections arecomputed. The elastic forces are calculated (Eq. 14) andused as external forces in the dynamic equations.

Fig. 7. Six link mechanism in its initial and final configuration

The kinematic parameters, the mass and stiffness

properties of mechanism links are as follows (all measures

are in SI UNITS): ternary link – size 1×1×1, mass 4m = 3,

inertia moment 4 J = 0.3; link 1 - length 1l = 1, 1m = 1,

1 J = 0.1; link 2 – =2l 2, =2m 2, =2 J 0.2; flexible links -

== 53 ll 2, mass density ρ = 3000, modulus of elasticity

111070 ×= . E , cross section area 4104 −×= A , second

moment of area 7102 −×= z I . The initial configuration of

the mechanism is defined by the coordinates 531 , ,i ,qi = as

it is shown in Fig. 7, i.e.: 1q = ; 23 π −=q ; 655 =q .

The prescribed motion is realized as a reonomic constraint

for 1q , i.e.: ⎟ ⎠

⎞⎜⎝

⎛ =

41

t cos.q for 40 ≤≤ t ; const q =1& for

t > 4.

0 X

0Y

1Y

1 X

1q2 X 2Y

4 X

4Y

3 X

5 X

5Y

3Y ' x f 3

" x f 3

' Y 3

" X 3

' X 3

" Y 3

' m3

" m3

' y f 3

" y f 3

flexible

link 3

1

24

3

5

' m

5

' Y 5

" X 5

' X 5 " Y 5 ' x f

5

" m5

' y f 5

" x f 5

" y f 5

flexible

link 5

Initial position t = 0 Final position t ≈ 8




The time histories of the mechanism motion, links 1, 3and 5, are presented in Fig. 8. The time histories of theelastic longitudinal forces in links 3 and 5 are shown inFig. 9 and 10. The longitudinal elastic forces that arise as

a result of the compulsive motion of the crank 1 do notcause significant influence of the transfer functions 3q

and 5q . The integration process starts from the initial

singular configuration and stops when the mechanism

reaches another singular configuration at t ≈8. It could beseen that longitudinal elastic forces are exaggerated in theinitial stage for going out from the singular configuration.These forces become higher at the end of motion sinceadditional inertia forces appear and become extremelyhigh at the interruption of the integration process.

The example demonstrates the applicability of the

numerical algorithm. Future investigations will include

damping and friction forces. Numerical integrationmethods for suppression of the high order vibrations will be applied and investigated.

V. Conclusions

An approach to simulation of rigid and flexiblemultibody system is proposed for which the kinematic

constraints are substituted by elastic forces in the flexiblelinks. The method provides general solution for everykind of closed chains including over-constrainedmechanisms and singular configurations.

Acknowledgement

The financial support of National Fund “ScientificInvestigations”, Bulgarian Ministry of Education andScience, MI – 1507/2005 is acknowledged.

References

[1] Schiehlen W., (ed.). Advanced Multibody System Dynamics. Solid

Mechanics and its Application. Kluwer Academic Publishers,

Dordrecht, 1993.

[2] Haug E. J., Computer-aided Kinematics and Dynamics of

Mechanical Systems. Allyn and Bacon, Boston, 1989.[3] Shabana A. A., Dynamics of Multibody Systems. John Wiley &

Sons, New York, 1989.

[4] García de Jalón, J. and Bayo E, Kinematic and Dynamic Simulation

of Multibody Systems. The Real-Time Challenge. Springer-Verlag,

New York, 1993.[5] Angeles J. and Kecskemethy A., Kinematics and Dynamics of

Multibody Systems. Springer-Verlag, Berlin, 1995.

[6] Bennet G. T., A new mechanism, Engineering, London, 76: 777 –

778, 1903.[7] Gear C. W., The simultaneous numerical solution of differential-

algebraic equations. IEEE Trans. Circuit Theory, CT-18:89—95,

1971.

[8] Baumgarte J., Stabilization of Constraints and Integrals of Motion.Computer Methods in Applied Mechanics and Engineering, (1):1-

16, 1972.[9] Kurdila A. J. and Narcovich F. J., Sufficient conditions for penalty

formulation method in analytical dynamics. Computational

Mechanics, 12:81 – 96, 1993[10] Bayo E. and Ledesma R., Augmented Lagrangian and mass-

orthogonal projection methods for constrained multibody dynamics. Nonlinear Dynamics, 9:113—130, 1996.

[11] Blajer W., Schiehlen W., and Schirm W.. A projective criterion to

the coordinate partitioning method for multibody dynamics. Applied

Mechanics, 64:86—98, 1994.

[12] Blajer W., A geometric unification of constrained system dynamics. Multibody System Dynamcis, 1:3—21, 1997.

[13] Wehage R. A. and Haug E. J., Generalized coordinate partitioning of

dimension reduction in analysis of constrained dynamic systems. ASME Journal of Mechanical Design, 104:247—255, 1982.

[14] Bayo E. and Garcia de Jalon J.. A modified Lagrangian formulation

for the dynamic analysis of constrained mechanical systems.Computer methods in applied mechanics and engineering, 71:183—

195, 1988.[15] J. Cuadrado J., Cardenal, and Bayo E., Modeling and solution

methods for efficient real-time simulation of multibody dynamics. Multibody System Dynamics, 1:259—280, 1997.

[16] Arabian A. and Wu F., An improved formulation for constrained

mechanical systems. Multibody System Dynamics, 2: 49-69, 1998.

[17] Eich-Soellner E. and Fuhrer C., Numerical Methods in Multibody

Dynamics, B.G. Teubner, Stuttgart, 1998.

[18] Aghili F. and Piedboeuf J.-C., Simulation of motion of constrainedmultibody systems based on projection operator, Multibody System

Dynamics, Kluwer Academic Publishers, 2002. [19] Zahariev E. and McPhee J., Stabilization of multiple constraints

using optimization and a pseudo-inverse matrix, Mathematical and

Computer Modeling of Dynamical Systems, Swets & Zeitlinger Publishers, Netherlands, 9(4):423 – 441, 2003.

[20] Zahariev E., Generalized finite element approach to dynamics

modeling of rigid and flexible systems, Mechanics Based Design of

Structures and Machines, 34(1): 81-110, 2006.

[21] Zienkevich O. C., T he Finite Elment Method . McGraw-Hill, 1997.

[22] Zahariev E., Relative finite element coordinates in multibody ystemsimulation, Multibody System Dynamics, Kluwer Academic

Publishers, 7:51 – 77, 2002.

-2

-1

0

1

2

3

4

0 2 4 6 8

5q

3q 1q

t

Fig. 8. Time history of the characteristics of motion of links 1, 3, 5

-20000

-15000

-10000

-5000

0

5000

10000

15000

20000

25000

0 2 4 6 8 t

Fig. 9. Time history of the longitudinal elastic forces in link 3

t -40000

-30000

-20000

-10000

0

10000

20000

30000

0 2 4 6 8

Fig. 10. Time history of the longitudinal elastic forces in link 5

Date post:	04-Apr-2018
Category:	Documents
Upload:	chernet-tuge
View:	217 times
Download:	0 times

2007 06 IFToMM Dynamics

Documents